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High Energy Physics - Theory

Title:
Integrability and cycles of deformed ${\cal N}=2$ gauge theory

Abstract: To analyse pure ${\cal N}=2$ $SU(2)$ gauge theory in the Nekrasov-Shatashvili
(NS) limit (or deformed Seiberg-Witten (SW)), we use the Ordinary Differential
Equation/Integrable Model (ODE/IM) correspondence, and in particular its
(broken) discrete symmetry in its extended version with {\it two} singular
irregular points. Actually, this symmetry appears to be 'manifestation' of the
spontaneously broken $\mathbb{Z}_2$ R-symmetry of the original gauge problem
and the two deformed SW cycles are simply connected to the Baxter's $T$ and $Q$
functions, respectively, of the Liouville conformal field theory at the
self-dual point. The liaison is realised via a second order differential
operator which is essentially the 'quantum' version of the square of the SW
differential. Moreover, the constraints imposed by the broken $\mathbb{Z}_2$
R-symmetry acting on the moduli space (Bilal-Ferrari equations) seem to have
their quantum counterpart in the $TQ$ and the $T$ periodicity relations, and
integrability yields also a useful Thermodynamic Bethe Ansatz (TBA) for the
cycles ($Y(\theta,\pm u)$ or their square roots, $Q(\theta,\pm u)$). A latere,
two efficient asymptotic expansion techniques are presented. Clearly, the whole
construction is extendable to gauge theories with matter and/or higher rank
groups.